# Dedekind modular law

Dedekind modular law. If $A,B,C$ are subgroups of a group $G$ with $A \subseteq B$ then $A(B \cap C) = B \cap AC$.

Below is what I want to prove. Let K be a finite group with $K = LH$, where $L,H$ are subgroups of $K$ with relatively prime orders. If $U$ is a maximal subgroup of $L$ then $UH = HU$. Proof:

$HU = HU \cap LH = (HU \cap L)H = (H \cap L)UH = UH$

Is my proof true?

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